In this note, we study automorphic forms and representations of . First, we describe local theory, archimedean and non-archimedean, and then global theory. This note is mainly a summary of a part of Bump’s Automorphic forms and representations (bu?), from chapter 2 to 4.

Introduction

Modular forms and Maass wave forms are certain functions defined on the complex upper half plane that satisfies -transformations laws (or more generally, transform under congruence subgroups ). There are a lot of applications of modular forms in number theory, such as sum of squares and the irrationality of , and the Wiles’ famous proof of Fermat’s Last Theorem. There are also applications in other subjects, such as combinatorics (partition numbers), physics, representation theory (monstrous moonshine), knot theory, etc.

In this note, we will study how to interpret such functions (so-called classical automorphic forms) as a representation of adéle groups (here is a ring of adéles of global fields such as ), and study representation theory of it. This can be a starting point of the Langlands’ Program, which connects representation of Galois groups, algebraic geometry, and automorphic forms (representations).

To study such representations, we first study local representations. There are two kinds of local representations - archimedean and non-archimedean. For the archimedean cases, we study representation theory of via so-called ()-modules. -module is a vector space with compatible and -actions. It is easier to study -modules than studying the representation of directly since -modules are more algebraic. We will classify -modules for and also study which of them are unitarizable, since we are interested in the representation that lives in space. Also, we will see how these representations are related to classical automorphic forms (such as modular forms and Maass wave forms).

We also have non-archimedean representations - which are representation of -adic groups for a prime . They are very different from archimedean cases because of their topology. This makes the situation easier or harder, but anyway, we will also classify all the representations of such groups and study their unitarizability.

When we finish the local theories, we can glue these representations to obtain the representation of the adéle group . (In fact, this is not a true representation of , but a representation of .) While we are studying such representations (local or global), we will only concentrate on some nice representations (admissible representations) that are close to the representation of finite groups. Automorphic representations are some nice representations that also satisfies some analytic conditions on growth. Later, we will see that Flath’s decomposition theorem tells us that it is enough to study such glued representations to study automorphic representations.

Before we get into the representation theory of , we will study first, which are completed by Tate in his celebrated thesis. He find a natural way to prove the analytic continuation and the functional equation of Hecke’s -function using local-global principle, and such idea will be used to define -functions attached to automorphic representations of .

It may be hard to study an abstract representation of a given group (such as or ). Whittaker model (or Whittaker functional) help us to study such representations as a very concrete representation that functions on the group lives (and the group acts as a right translation). Most case, such Whittaker model exist and unique, and such results are called (local or global) multiplicity one theorem. In the last section, we will see how the multiplicity one theorem is related to the classical modular forms.

Archimedean theory

In this section, we will study representation theory of the group . Usually, it is easier to study representation of compact groups than non-compact groups because it is not much different from the representation theory of finite groups. First, any finite dimensional representations are unitarizable, by taking average of arbitrary hermitian inner product on the space over all group with respect to Haar measure, which is finite for compact groups. Also, we have celebrated Peter-Weyl theorem, which claims that any unitary representation (including infinite dimensional representation) on a complex Hilbert space is semisimple, i.e. can be decomposed as a direct sum of irreducible dimensional unitary representations, and these are all finite dimensional and mutually orthogonal. It is also known that representation of compact group are completely determined by its character.

Also, Lie algebra representations of (or its complexification ) are much easier than studying the representation of Lie group, because it is a linearlized version of original representation and we have a lot of tools to use. We even have a complete classification of semisimple Lie algebra over , which is a very rich theory itself.

Instead of studying representations of directly, we will study representation theory of its maximal compact group and Lie algebra representation of . Eventually, we will consider so-called -module, which is a vector space with compatible actions of and , and the space is not so big to deal with, i.e. admissible. We give complete classification of -module for and , and investigate which of them are unitarizable. Since unitary representation of is completely determined by associated -module, we also get a complete classification of unitary representations.

In the last subsection, we will also see how the representation theory of can be used to study spectral problems (of classical automorphic forms).

Representation theory of

Geometrically, Lie algebra of a Lie group is a tangent space at the identity, and it has a structure of Lie algebra given by a Lie bracket. In case of and , their Lie algebra is , the space of real matrices with the Lie bracket . The most important point is that any representation of Lie group induces a Lie algebra representation.

The finite dimensionality assumption is non really necessary. In fact, we will only consider special kind of representation: right regular representation on . The statement is also true for this case, even if the space is not finite dimensional.

By the universal property of universal enveloping algebra , any Lie algebra representation can be extended to a representation of . We will regard as a ring of differential operators, which are left-invariant since Lie algebra action is obtained by differentiating right regular representation. When the element is in the center of the universal enveloping algebra , it is both invariant under the left and right regular representations.

Now we will concentrate on . is generated by the elements with relations Now let be an element in , where the multiplication is in , not a matrix multiplication. This is a very special element in , which is called the Cacimir element. The element is in the center of , and in fact the center is generated by and .

We will consider the complexification of and slightly modify the elements in as Then they satisfy the same relations as , and . Indeed, we have where is the Cayley transform. We will see the reason why we are using instead of , and , in section 2.6.

For an arbitrary representation of , there may not exists a corresponding Lie algebra action on since the limit may not exists. We will define as a largest subspace where such action exists, i.e. the limit exists for all and . We will call such as smooth vector, and we can easily check that such space is invariant under the action of from the equation Also, the action of on is a Lie algebra representation. We define the action of on as for . We can show that the subspace of smooth vectors is not so small, indeed, it is dense in .

Representation theory of compact group

In this section, we will see how representations of compact groups well-behaves. We will prove the Peter-Weyl theorem, which claims that every representation of a compact group decomposes as a direct sum of finite dimensional irreducible representations.

For any finite group and it’s irreducible representation (which has finite degree), we can construct a -invariant inner product on : choose any inner product and define a new pairing as Then this pairing is also an inner product on and it is -invariant by definition. We can do the same thing for a representation of compact group on a Hilbert space , by integrating a given inner product on over with respect to its Haar measure. (Note that compact group has a finite Haar measure.) This induces same topology as before.

Now we will prove the most important theorem in the representation theory of compact groups, Peter-Weyl theorem. For a representation on a Hilbert space of , a matrix coefficient of the representation is a function on of the form . We need the following proposition:

Using Peter-Weyl theorem, we can define admissibility of representation of for or . A representation of is admissible if each isomorphism class of finite dimensional representations of occurs only finitely many times in a decomposition of . This implies that for each irreducible representation of , the isotypic component of , the direct sum of all the subrepresentations of isomorphic to , is finite dimensional. We can check that multiplicity of a given finite dimensional representation does not depend on the decomposition. Also, it is a right category to study since it is known that any irreducible unitary representation is admissible.

The next result shows that in the decomposition of irreducible admissible unitary representation over , the multiplicity of the trivial representation of is at most one. To prove this, we need the result about commutativity of Hecke algebra which can be proved by Gelfand’s trick with Cartan decomposition.

Note that is non-commutative.

For , we can prove a similar result when we consider the subalgebra of where acts as a nontrivial character , i.e. . Let be a subalgebra of such functions.

Now we can prove the uniqueness of the -fixed vector.

Note that the admissibility condition is unnecessary because any irreducible unitary representation is admissible (as we mentioned above).

-module for and classification

Now we can define the -module, which is a thing what we really want to study. In some sense, the subspace of smooth vectors is still too big to study. We will consider much smaller space, the space of -finite vectors , which is also dense in but much easier to study algebraically.

Motivated by this, we define a notion of -module, which is a vector space of -finite vectors with compatible actions.

For example, if is an admissible representation of , then is a -module. We will classify all the irreducible admissible -module for . First, we will do for with , and modify it to get the result for with .

Let be a irreducible admissible -module, so that it can be decomposed as an algebraic sum of isotypic parts where each is finite dimensional. Since is abelian, all the irreducible representations are 1-dimensional, and they are parametrized by integers as . Hence we can write as where . Each is at most 1-dimensional by Theorem . Now we can extend the -action to -action naturally. The set is called the set of -types. We have the following Schur’s lemma for -modules.

This proposition shows that the elements acts as scalars on . (This will be the parameter to classify -modules later.) The following proposition gives a description how the elements acts .

By the proposition, we have that the set of -types of is all even or all odd. This defines a parity of , even or odd. The following theorem tells us the uniqueness of representations, whether we don’t know the existence yet.

Now we will give a construction of such representation with given parameters, which will finish the classification. Let or , which represents parity of a representation, and let be two complex numbers. Let and , where . As you expect, these will be scalars corresponding to and .

Note that the right translation action (regular action) extends to . We can also prove that is the space of smooth vectors for this representation. By Iwasawa decomposition, we have for , so each is determined by its value on , and can be any smooth function, subject to the condition .

In fact, the representation is an example of an example of induced representation. For a locally compact Hausdorff group and its subgroup , we can obtain a representation of from a representation of in a canonical way: if is a representation of , then define where are modular characters. If we give -action on by right translation, then this gives a representation of . We denote such representation by . Now let , (the subgroup of upper triangular matrices in ), and let be a character defined as Then, by definition, the representation is just . Note that is a unimodular group (so that is trivial) and

Now we want to study -module of -finite vectors in . If , then there exists a unique such that , which satisfies . Iwasawa decomposition gives an explicit description of : By the direct computation, we can show that this function satisfies the following relations:

Now, as you expect, these representations give examples of the previous representations with two parameters and -type. The above ’s generate the space of -finite vectors.

In other words, this gives a classification of -module for .

When is not of the form , then the equivalence class of irreducible admissible -modules of with given are denoted by . When , we denote it as and it is called principal series. (By tensoring with a suitable power of determinant, we can assume easily.) Later, we wiil check that the representation is unitarizable if and only if and , so we will concentrate on this case.

The finite dimensional representation with a set of -types can be realized as a space of polynomials: consider the space of homogeneous polynomials of degree in two variables , and let acts on the space by which is a degree irreducible admissible representation where acts as the scalar . This will not appear again since it is not unitarizable. (We will prove that the only finite dimensional unitarizable representation is 1-dimensional, which factors through the determinant map.)

If , we have irreducible admissible representations with set of -types as , and equivalence class of these representations will be denoted as and called the discrete series. When , the representations are called limit of discrete series.

To classify -modules of , we need some modification. We can check that the representation of has a symmetric property: the set of -types is symmetric so that if and only if . Hence cannot be extended to , but can be. We will denote the latter one by , with -module structure.

For the construction of principal series reresentation of , we define as for , where and . Then we denote as , and we will denote by the underlying -module of -finite vectors. Note that since each function in is determined by its restriction to . So there are two extensions of the -module structure on to a -module structure (corresponds to the choice of ), and the same is true for the corresponding -modules.

Unitaricity and intertwining integrals

Now we will see which representations in the above list are unitarizable. For some special case (so-called complementary series), we will show that the representation is unitary by using the intertwining integral, which is an hidden explicit isomorphism between two isomorphic -modules.

The following theorem tells us that induced representation of unitary representation is again unitary in some special case.

Using this, we can prove that there are some class of representations that are induced from unitary representation, so is unitarizable.

If is a unitary representation of and , then the action of on is skew-symmetric, i.e. for all . Especially, we have . The following theorem give some necessary conditions for unitaricity.

From the above theorems, we know unitarizability of except for and . We will also show that these representations are also unitary, but induced from nonunitary representations of Borel subgroup. Such representations are called complementary series, and the corresponding eigenvalues are exceptional eigenvalues.

To construct such representation, we will use intertwining integral. We know that and are isomorphic, when they are irreducible, as -module since they have the same and . Also, when they are not irreducible (when ) they are not isomorphic, but their composition factors are isomorphic. We will construct an intertwining map between those to representations as an integral.

For